[From Bill Powers (2011.07.05.0225 MDT)]
Rick Marken (2011.07.04.2200) –
RM: If this pans out, doesn’t it
suggest that the dynamics of the filtered output of the open-loop
model is not equivalent to that of the control model?
BP: That’s not because it couldn’t be equivalent, but only because you
haven’t set it up to be equivalent. The point is that in order to do
this, you have to look at how the closed-loop system works with a given
environment, and then duplicate that environment exactly for the
open-loop system. Then you can find an open-loop transfer function,
output as a function of disturbance, that exactly imitates the behavior
of the closed-loop system. In other words, design the open-loop system to
recreate the behavioral illusion as closely as possible. The fit can be
perfect, and maybe the open-loop system can even perform a little better
– but only if the environment remains exactly the same, with no
unpredictable kinds of disturbances and no changes in physical
characteristics. Also, a reference signal will be needed to set the level
of input at which the output is just zero. That has to stay constant,
too. In S-R theory that’s just the intercept of the regression line,
isn’t it?
Come to think of it, the open-loop model can’t perform “better”
than the open-loop model, because there’s no a priori way to define what
“better” means. The open-loop model can be adjusted so it fits
any specific example of real behavior as well as a closed-loop model fits
it. But if you don’t know how the closed-loop model would work, you
aren’t trying to minimize error so you just look for what seems to be a
good fit. You can set your standards for “good” any way you
like.
This is basically why the stimulus-response model has lasted as long as
it has. In a fixed environment one can match an S-R model to what the
real system is doing, but if and only if nothing unpredicted ever
happens. If that weren’t true, the S-R model would have obviously failed.
But it doesn’t fail if you use highly controlled conditions, and if you
don’t ask that the model work in any great detail. If all you ask is that
the model behave in a way that has modest qualitative correlations with
observation (that is, things happen or don’t happen, without paying too
much attention to how much of anything happens), some of the time,
under some conditions, then you can build up stories about how difficult
it is to understand behavior and how wonderful it is that we can get the
little scraps of knowledge that we have managed to find.
RM: After all the control
model does have three levels and there is a disturbance to the
perception controlled by the top level (the logical relationship between
output and cursor color).
BP: All of that can be transformed into an input-output transfer
function. Just solve the equations of the control system for output as a
function of disturbance and reference signal, then make sure that only
known and predictable disturbances take place, that the reference signal
never changes, and that the output function of the system doesn’t
change, either. The control system doesn’t need to be protected that way,
but the open-loop system does. That’s the basis of our whole strategy for
disproving S-R theory. We let those factors vary (all but the reference
signal, the intercept of the regression line) and show that the control
system model and the real system go on behaving in nearly the same way,
and the S-R model doesn’t.
Keep in mind that the object of this strategy is to disprove the theories
people have believed in, not merely to dispute them. Since the people we
are trying to persuade are just as smart as we are, we have to assume
that there were good reasons for those beliefs and reasonable
demonstrations that they were correct (allowing for low standards of
proof adopted because higher standards simply couldn’t be met). If they
had known about control systems, so as to have some other interpretation
with which to compare the one they were using, they would have been just
as able as we are to see that a change was necessary, given that pride
didn’t interfere. So we aren’t arguing with anyone. We’re merely
providing the means by which they can convince themselves that their
theories are wrong, if they want to.
Best,
Bill P.